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## Filter estimation

The PEFs I estimate are time domain non-stationary filters to cope with the variability of seismic data with time and offset. I describe in the appendix how the PEFs are estimated. From the fitting goals in equation (32), we have
 (8)
where is a matrix for non-stationary combination, is a masking operator, a vector of the unknown PEFs coefficients, the data vector from which we want to estimate the PEFs and a regularization operator. I describe each of these elements in more detail in the appendix.

Often with seismic data, the amplitude varies across offset and time. These amplitude variations can be troublesome when we want to use least-squares inversion because they tend to bias the final result Claerbout (1992). Therefore, it is important to make sure that the amplitude variation does not affect our processing. One solution is to apply a weight to the data like Amplitude Gain Control (AGC) or a geometrical spreading correction. However, a better way is to incorporate the weight inside our inversion by weighting the residual Guitton (2003). Introducing a weighting function in the PEFs estimation, we have the following fitting goals:
 (9)
As shown by Guitton (2003), this weighting improves the signal/noise separation results. The choice of the weighting function will be discussed later for the multiple attenuation example. This weight can also be different for the noise and signal PEFs. If we want to find in a least-squares sense, we have to minimize the objective function
 (10)
which leads to the least-squares estimate of
 (11)
Because we have many filter coefficients to estimate, is estimated iteratively with a conjugate-gradient method.

Now, prior to the signal estimation in equation (5), and need to be computed from a signal and noise model, respectively. The multiple model is derived by autoconvolving the recorded data Verschhur et al. (1992). I then obtain a prestack model of the multiples that I use to estimate the bank of non-stationary PEFs .

It is important to keep in mind that at this stage, I assume that the relative amplitude of all order of multiples is preserved. In theory, an accurate surface-related multiple model can be derived if (1) the source wavelet is known, (2) the surface coverage is big enough, and (3) all the terms of the Taylor series that model different orders of multiples are incorporated Verschhur et al. (1992). In practice, however, a single convolution is performed (first term of the Taylor series) which leaves us with a multiple model with erroneous relative amplitude for high order multiples. In addition, the surface coverage might not be sufficient. This leaves us with wrong amplitudes for short offset traces and complex structures. Because PEFs estimate patterns, wrong relative amplitude can affect our noise estimation. However, as we shall see later, 3-D filters seem to better cope with noise modeling inadequacies.

The signal PEFs are more difficult to estimate since the signal is usually unknown. However, Spitz (1999) shows that for uncorrelated signal and noise, the signal PEFs can be expressed in terms of 2 PEFs: the PEFs , estimated from the data , and the PEF , estimated from the noise model such that
 (12)
Equation (12) states that the signal PEFs equal the data PEFs deconvolved by the noise PEFs. This deconvolution insures that the PEFs and will not span similar components of the data space. To avoid the deconvolution step suggested in equation (12), I apply the noise PEFs to the data vector and estimate the signal PEFs from the convolution result. This will give me an approximation of that I later use for the noise attenuation (Spitz, 2001, personal communication).

Thanks to the Helix Claerbout (1998); Mersereau and Dudgeon (1974), the PEFs can have any dimension. In this paper, I use 2-D and 3-D filters and demonstrate that 3-D filters lead to the best noise attenuation result. When 2-D filters are used, the multiple attenuation is performed for one shot gather at a time. When 3-D filters are used, the multiple attenuation is performed for one macro-gather at a time. A macro-gather is a cube made of fifty consecutive shots with all the offsets and time samples. When the multiple attenuation is done, the macro-patches are reassembled to form the final result. Note that there is an overlap of five shots between successive macro-gathers. In the next section, I show a prestack multiple attenuation example with field data from the Gulf of Mexico.

Next: A 2-D field data Up: Theory of multiple attenuation Previous: Multiple attenuation
Stanford Exploration Project
7/8/2003